As a numerical calculation method of calculating the motion of a continuum such as a fluid or an elastic body, for example, a finite difference method, a finite element method, or a finite volume method has been used which finds the approximate solution of a differential equation on the basis of the numerical mesh. In addition, in recent years, since numerical calculation has been used in the field of application such as computer aided engineering (CAE), the numerical calculation method of calculating the state of the continuum has been developed and the problem of the interaction between a fluid and a structure has been solved. However, in the numerical calculation method using the numerical mesh, when a moving boundary problem, such as the existence of an interface including a free surface or a problem in fluid-structure interaction analysis for analyzing the interaction between a fluid and a structure, occurs, handling of the continuum becomes complicated. Therefore, in some cases, it is difficult to create a program.
As the numerical calculation method without using the numerical mesh, there is a particle method. The particle method analyzes the motion of a continuum as the motion of a finite number of particles. A representative particle method which is currently proposed is, for example, a smoothed particles hydrodynamics (SPH) method or a moving particles semi-implicit (MPS) method. The particle method can analyze the motion of the continuum without a special measure in the treatment of the moving boundary. Therefore, in recent years, the particle method has been widely used as the numerical calculation method of calculating the motion of the continuum.
In particular, in the pressing of metal such as casting or forging, metal is processed through a complicated process. For example, metal (solidified metal) which is cooled and solidified is mixed with liquid metal, the solidified metal is grown, and the volume of metal is changed in the solidification process. The particle method is expected to be actively used in casting and forging simulations since the particle method has the advantage that it is easy to treat the free surface, it is relatively easy to calculate a parallel performance and interaction with a solid, or the like.
Cleary method has been known as a method of calculating a process (solidification process) in which a liquid is cooled and solidified, which is a basic technique for simulating a casting process. The Cleary method calculates the time evolution of the internal energy of each liquid particle using the SPH method which is one of the particle methods and calculates the temperature, density, and viscosity coefficient of the liquid particle as a function of internal energy. That is, when the internal energy is reduced and the temperature is lowered, the Cleary method increases the viscosity coefficient of the liquid to represent solidification and increases the density of the liquid to represent a reduction in volume due to solidification.
The Cleary method discretizes the equation of a fluid using the SPH method as represented by the following Expressions (1) to (4):
                                          ⅆ                          ρ              i                                            ⅆ            t                          =                              ∑            j                                                          ⁢                                                    m                j                            ⁡                              (                                                      v                    i                                    -                                      v                    j                                                  )                                      ·                                          ∂                                  W                  ⁡                                      (                                                                                                                  x                          i                                                -                                                  x                          j                                                                                                            )                                                                              ∂                                  x                  i                                                                                        (        1        )                                                      ⅆ                          v              i                                            ⅆ            t                          =                  g          -                                    ∑              j                                                                    ⁢                                                            m                  j                                ⁡                                  [                                                            (                                                                                                    ρ                            j                                                    +                                                      ρ                            i                                                                                                                                ρ                            j                                                    ⁢                                                      ρ                            i                                                                                              )                                        -                                                                  ξ                                                                              ρ                            j                                                    ⁢                                                      ρ                            i                                                                                              ⁢                                                                        4                          ⁢                                                      μ                            i                                                    ⁢                                                      μ                            j                                                                                                    (                                                                                    μ                              i                                                        +                                                          μ                              j                                                                                )                                                                    ⁢                                                                                                    v                            ij                                                    ·                                                      x                            ij                                                                                                                                                                                                                            x                                ij                                                                                                                    2                                                    +                                                      η                            2                                                                                                                                ]                                            ⁢                                                ∂                                      W                    ⁡                                          (                                                                                                                            x                            i                                                    -                                                      x                            j                                                                                                                      )                                                                                        ∂                                      x                    i                                                                                                          (        2        )                                          p          i                =                              P            0                    ⁡                      [                                                            (                                                            ρ                      i                                                              ρ                                              s                        ,                        i                                                                              )                                γ                            -              1                        ]                                              (        3        )                                                      ⅆ                          U              i                                            ⅆ            t                          =                              ∑            j                                                          ⁢                                                    4                ⁢                                  m                  j                                                                              ρ                  j                                ⁢                                  ρ                  i                                                      ⁢                                                            k                  i                                ⁢                                  k                  j                                                            (                                                      k                    i                                    +                                      k                    j                                                  )                                      ⁢                                                            x                  ij                                                                                                                                          x                        ij                                                                                    2                                    +                                      η                    2                                                              ·                                                ∂                                      W                    ⁡                                          (                                                                                                                            x                            i                                                    -                                                      x                            j                                                                                                                      )                                                                                        ∂                                      x                    i                                                                                                          (        4        )            
Expression (1) indicates the law of conservation of mass, Expression (2) indicates the law of conservation of momentum, Expression (3) indicates a state equation, and Expression (4) indicates the law of conservation of energy. In Expressions (1) to (4), xi, vi, ρi, mi, pi, and Ui are the position vector of a particle i, the velocity vector of the particle i, the density of the particle i, the mass of the particle i, the pressure of the particle i, and the internal energy of the particle i, respectively. In addition, xij and vij are the relative position vector and relative velocity vector of particles i and j, respectively, and xij=xi−xj and vij=vi−vj are established. Furthermore, κi and μi are the thermal conductivity of the particle i and the viscosity coefficient of the particle i, respectively. In addition, P0=ρ0c2 is established and c is the speed of sound. Further, ρs,i is the reference density of the particle i and pressure is 0 when ρi=ρs,i is established.
In addition, W is a kernel function and, for example, a spline function represented by the following Expression (5) is used as W.
                              W          ⁡                      (                          r              ,              h                        )                          =                  {                                                                                          (                                          1                      -                                              1.5                        ⁢                                                                              (                                                          r                              h                                                        )                                                    2                                                                    +                                              0.75                        ⁢                                                                              (                                                          r                              h                                                        )                                                    3                                                                                      )                                    /                  β                                                                                                  0                    ≤                                          r                      h                                        ≤                    1                                    ,                                                                                                      0.25                  ⁢                                                                                    (                                                  2                          -                                                      r                            h                                                                          )                                            3                                        /                    β                                                                                                                    1                    ≤                                          r                      h                                        ≤                    2                                    ,                                                                                    0                                                              2                  ≤                                                            r                      h                                        .                                                                                                          (        5        )            
In Expression (5), h is an influence radius between particles. For example, as h, a value that is about two to three times that the average distance between the particles in the initial state is used. In addition, β is a value which is adjusted such that the entire space integration amount of the kernel function is 1. In the case of two dimensions, β is set to 0.7 πh2. In the case of three dimensions, β is set to πh3.
In the Cleary method, when the internal energy is reduced and the temperature is lower than a melting point, the viscosity coefficient μi is increased and the effect of canceling the relative velocity between the particles represented by the third term of Expression (2) is improved. Therefore, it is difficult to deform by the third term. In this way, the Cleary method represents solidification. In addition, in the Cleary method, when the reference density ρs,i increases, pressure is reduced and the surrounding particles are collected by the effect of the second term of Expression (2). In this way, the Cleary method represents contraction due to solidification.
It is possible to perform a simulation by calculating the time evolution of Expressions (1) to (4) using the Euler's method or the Leapfrog method which is a general ordinary differential equation.
In the Cleary method, since the value of the viscosity coefficient increases in the solidification process, a time step is very small in calculation. Therefore, the number of calculation operations increases until calculation ends. As a result, the Cleary method has a long calculation time.
As an example of a method of calculating the interaction between a fluid and a rigid body, there is a method which uses the equation of motion of a liquid for a liquid portion and uses the equation of motion of a rigid body for a solid portion. In the method, since the motion of the solid portion is calculated by the equation of motion of a rigid body, the calculation time is shorter than that in the Cleary method.
As to the conventional techniques, refer to Paul W. Cleary, “Extension of SPH to predict feeding freezing and defect creation in low pressure die casting”, Applied Mathematical Modeling, 34 (2010), pp. 3189-3201; and Koshizuka, S., Nobe A. and Oka Y. “Numerical Analysis of Breaking Waves Using the Moving Particle Semi-implicit Method”, Int. J. Numer. Meth. Fluids, 26, 751-769 (1998), for example.
However, in the method which uses the equation of motion of a rigid body for the solid portion, the accuracy of the calculation result is not high in a situation in which a new solid is generated from a liquid. For example, a case in which liquid metal is poured into a mold and then cooled will be described. In this case, a plurality of portions of the liquid metal starts to be solidified depending on the cooling conditions and the volume of the plurality of solidified portions increases over time. Then, the entire liquid metal is solidified. FIG. 14 is a diagram illustrating an example of the problems of the method according to the related art. In the example illustrated in FIG. 14, particles 90a of a solid portion 90, particles 91a of a solid portion 91, and particles 92a of a liquid portion 92 in the metal which is solidified by cooling are present in a mold. In this case, even when the solidified volume of the solid portion 90 is increased, the liquid portion 92 is solidified, and the solidified portion 92 and the solid portion 90 form the same solid by cooling, the above-mentioned method treats the solidified portion 92 and the solid portion 90 as individual solids. That is, the above-mentioned method separately calculates the motion of the solidified portion 92 and the motion of the solid portion 90 using the equation of motion of a rigid body. Therefore, the above-mentioned method separately calculates the motions of a plurality of solids even though there is originally one solid. As a result, the accuracy of the calculation result is not high.